There are many different types of telescope. The following is a brief overview of the main types of telescope. It is by no means an exhaustive list, but should help clarify the differences between the main configurations
Refractors are what one thinks of as the classical telescope. They use an objective lens to create an image of distant objects (such as a planet) and an eyepiece to magnify and view that image. The objective lens is usually designed to reduce the amount of chromatic aberration (false colour - caused by the lens acting like a prism and dispersing the light). An achromatic refractor reduces, but does not entirely eliminate the fringes of false colour that can be seen. An apochromatic refractor uses exotic (and very expensive) glass to virtually eliminate any chromatic aberration, except at exceptionally high magnification on very bright objects. Until recently, apochromats were much more expensive than achromats. There are now a few apochromats on the market that are comperable in price to similar, high-end achromats.
Reflecting telescopes use a concave mirror to focus the light from an object, rather than using a lens as the refractors do. An eyepiece lens is still used to magnify and observe the resulting image.
The two main configurations of reflector are the Newtonian and the Cassegrain. In the Newtonian design a small secondary mirror is mounted at 45 degrees, diverting the light to the side of the tube for viewing. In the Cassegrain, a convex secondary reflects the light through a hole in the primary mirror for viewing.
"Cats" are scopes that use a combination of lenses and mirrors to produce the primary image. The two main cats are the Schmidt-Cassegrain and the Maksutov-Cassegrain. The Schmidt-Cassegrain uses a thin, aspheric front lens called a "corrector plate" to reduce image deffects produced by the two mirrors. The Maksutov is similar, but uses a thick, spherical meniscus lens to correct the image.
Some less common variants are the Maksutov-Newtonian, which uses a thick meniscus lens, but has a flat secondary angled as the Newtonian's, and the Kletsov-Cassegrain, which has no front corrector lens, but instead has a small corrector lens placed in front of the surface of the small secondary mirror.
One type of scope that is marketed as a catadioptric design, but that should generally be avoided, is a short-tube newtonian with a small lens mounted at the end of the focuser. In general, these are lower quality telescopes. Here's a hint: if it says it is a Newtonian telescope with a 1m focal length, but the tube is only 50cm long, that should be a warning. If it claims to be an "innovative catadioptric design", just back away slowly.
Any time you see a telescope marketed as "600 Power professional telescope!", just laugh at it. Ignore the Hubble photos on the side of the box. They were not taken with that scope, I promise. To understand the capabilities of a telescope, you need to understand the meanings of aperture, focal length, and focal ratio.
The aperture of a telescope is it's ability to gather light. Basically, it is just the clear diameter of the objective element (primary mirror in a reflector, objective lens in a refractor, corrector lens of a catadioptric). The greater the aperure, the more light it collects, and the brighter the image will be. Since much of what you do in astronomy is to view very faint objects, a large aperture is needed to make these objects more visible. A large aperture also increases the resolution of the telescope, meaning it increases the amount of detail that can be seen on a clear night. A third benefit, partly related to the second, is that a large aperture reduces the amount of diffraction around a bright object, producing tighter images.
The focal length of the scope is the effective distance from the primary mirror or objective lens, to the image it forms. The longer the focal length, the greater the intrinsic magnification of the telescope. Note, however, that the longer the focal length, the narrower the field of view, in general. So a long focal length is good for planetary observing, but may not be the best for open clusters and other large objects.
The focal ratio of a telescope is its focal length divided by its aperture. Thus, a scope with a focal length of 1200mm and an aperture of 200mm has a focal ratio of 6, designated as f/6. This is the same as the f-stop on a camera. A longer f-ratio for any given aperture means a greater magnification but a narrower field of view. Smaller f-ratios are also better for photography.
The magnification of a telescope depends on which eyepiece you use. Eyepieces all have a focal length of their own, measured in millimetres. The magnification or "power" of a telescope can be calculated by dividing the focal length of the telescope by the focal length of the eyepiece. For example, a 25mm eyepiece in a scope with a 1200mm focal length will give a magnification of 48x. A 10mm EP in the same telescope will give 120x.
Reflector telescopes (Newtonians, Cassegrains, or any of the many variations) have a secondary mirror which blocks a portion of the incoming light. As a result, the light seen by the scope is ring-shaped, rather than disk shaped. Having a central obstruction can reduce the contrast and increase diffraction, as well as reduce the actual amount of light hitting your eye. This latter effect is actually quite small - an obstruction of 20% of the diameter only blocks 4% of the incoming light.
The exit pupil is the diameter of the beam of light leaving the eyepiece and entering your eye. A fully dark adapted eye can have a pupil diameter of 7mm, but this maximum usually drops to about 5mm in older adults. As a result, an exit pupil of more than 5-7mm will result in some loss of brightness, as part of the beam will hit the iris of the eye, rather than the retina. This is not necessarily bad, as one can still enjoy a wide, rich field of view even with some light loss. At the other extreme, exit pupils much smaller than 1mm can become uncomfortable, in that it may be difficult to centre your eye over the eyepiece properly, and "floaters" in the eye can be more obtrusive. That being said, I have on occasion observed with a 0.5mm exit pupil without difficulty. Practice helps.
In order to calculate the exit pupil for any given scope/EP combination, divide the focal length of the eyepiece (in mm) by the focal ratio of your scope. A 12mm eyepiece in an f/6 scope will give a 2mm exit pupil.
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